On certain identity related to Jordan *-derivations

In this paper we prove the following result. Let H be a real or complex Hilbert space, let L(H) be the algebra of all bounded linear operators on H and let A(H) ⊆ L(H) be a standard operator algebra. Suppose we have an additive mapping D:A(H) → L(H) satisfying the relation D(An)=D(A)A* n-1+AD(An-2)A* +An-1D(A) for all A A(H) and some fixed integer n>1. In this case there exists a unique B L(H) such that D(A)=BA*-AB holds for all A A(H)